Hydrodynamic (HD) Simulation of N-Channel MOSFET’s with a Computationally Efficient Inversion Layer Quantization Model

A quantum mechanical treatment of electron inversion layers is incorporated in the hydrodynamic (HD) transport model used in UT-MiniMOS. A physically based, yet computationally efficient, three-subband model is implemented in the HD simulation tool. The three-subband model, which is based upon solutions to Schrodinger’s equation, has the important advantage of more accurately predicting the distribution of electrons in the inversion layers than does more conventional classical models. A more simplified quantum mechanical model with carrier heating effects included has also been developed. Terminal currents are calculated using these quantum mechanical models and the comparison with results from classical calculations indicates the importance of quantum mechanical effects in the deep submicron device simulations.


INTRODUCTION
The quantum mechanical treatment of carrier confinement in silicon MOSFET inversion layers is increasingly essential for accurate simulation of devices scaled into the deep submicron region. It is well known that a high transverse electric field at the Si/SiO2 interface leads to a large energy band bending, which in turn can cause quantization of the inversion layer carriers in the direction per-pendicular to the interface [1]. Self-consistent solutions of the Schr6dinger's and Poisson equations have been adopted by a number of researchers to investigate the two-dimensional nature of electrons in inversion layers [2,3]. However, from the point of view of the device design engineer, less computationally expensive approaches that may be routinely used with existing device simulators are necessary. Van Dort's simple quantum model [4] and three subband quantization model (3SB) [5] have been implemented in drift-diffusion simulation tools and have facilitated the prediction of device electrical behavior (such as threshold voltage and capacitance). The three subband model introduces an additional level of physical sophistication by solving for the energy-level and wave-function for the first three subbands, using approximate solutions to Schr6dinger's equation based upon physically reasonable estimates of the form of the potential well. This approach has the advantage of correctly predicting the electron distribution in the inversion layers due to quantum mechanical effects and, in turn, leads to better predictive capabilities for C-V simulation and physical oxide thickness estimation. In the hydrodynamic (HD) 3-subband simulation, carrier temperature instead of lattice temperature is used to distribute electrons among each subband and between the classical (three dimensional electron gas) and the quantum (two dimensional electron gas) domains. As a result, under high drain bias condition, a large portion of electrons near the drain edge will be in the classical domain due to the carrier heating effect. In this paper, we present, for the first time, HD simulation results with a 3SB model included using UT-MiniMOS [6].

CARRIER CONFINEMENT AND QUANTIZATION MODEL IMPLEMENTATION
When carriers are quantized in the potential well, the density of states must be represented by a twodimensional (2-D) density of states instead of the three-dimensional (3-D)  Schr6dinger's equation is the Airy function where the eigenenergies for the system can be calculated exactly as a function of the surface field [2,5]. To account for the perturbation of the potential due to the inversion layer charges, a variational wavefunction originally proposed by Fang and Howard [7] was used for the lowest subband. This bypasses the computational expense of solving Schr6dinger's equation explicitly. In order to maintain additional computational efficiency, only the first three subbands are used in the calculation of (1). Above the energy level Eel, the carriers are all treated as classical carriers with a 3-D density of states. This approach ensures agreement with classical calculations in the limit of low channel doping in addition to reducing the total computation time.
Since the electron temperature (or average energy) is used as an additional variable in the HD simulation, the temperature effect must be included in the quasi-Fermi level calculation in order to obtain a correct carrier distribution among subbands. Also, due to the electron heating, the classical carrier component may play an important role in the carrier density calculation for the region near the drain edge. The quasi-Fermi level and the classical carrier component need to be calculated correctly in the 3SB model to ensure that the HD simulation converges and that the simulation results are physically correct. In our approach, the fractional electron population in the quantum domain as a function of the electron energy is modeled using our quantum-window enhanced uniform-field Monte Carlo (MC) simulator [8], as shown in Figure 1. The fractional population-energy relationship in our HD simulation is obtained from Monte Carlo simulation under the low transverse effective field condition (Etrans--1.9 10 5 V/cm  effective density of states. The carrier distribution along the depth direction is subsequently calculated in the 3SB model to obtain the effective intrinsic carrier density, following the similar approach of Ref. [5]. In addition to the more physically accurate three-subband model, more simplified models are of interest to the modeling community. While these models do not attempt to predict the quantum mechanically consistent charge distribution, they are able to predict the effect of quantization on the terminal characteristics when this is the only information required. This is accomplished by altering the bandgap of the silicon near the interface via the intrinsic carrier concentration [4]. The change in the bandgap is modeled as follows: Esi ) ,2/3 /X EG t 4qkTcarrier E (2) where Es denotes the transverse electric field and/3 the fitting parameter used to calibrate the model to experimental data and/or more physically based quantization models. This modified bandgap is then used to make corrections to the charge density near the interface. The carrier temperature is used when the HD models are turned on, otherwise the lattice temperature is used. This modification permits the model to approach the classical results when the energy distribution of the carriers is highly energetic, and hence more classical-li-ke.

SIMULATION RESULTS AND DISCUSSION
In order to explore the importance of including carrier confinement models, as well as carrier heating models, in device simulation tools, we have examined the behavior of the drift-diffusion and hydrodynamic transport models both with quantum mechanical carrier confinement models and without. We used as our prototypical device structure the simple 0.18 micron nMOS device presented in Figure 2. The doping profiles, junctions depths and oxide thickness were selected to be representative of devices of this technology generation.
An examination of the electron concentration near drain edge at low drain bias (Va 0.1 V) illustrates that the electron distribution obtained from the HD 3SB calculation is almost the same as that from a DD-3SB calculation (Fig. 3(a)). This result can be understood by the fact that there is almost no carrier heating for such a low drain bias.
At high drain bias (Va 1.8 V), on the other hand, the distribution near the drain edge obtained from the HD-3SB calculation differs significantly from that obtained with DD 3SB (Fig. 3(b)). In the drift-diffusion approach, regardless of lateral field conditions, the carriers are assumed to be at the lattice temperature, so the carriers are assumed to occupy the lowest subbands (i.e., quantum mechanical regime) more readily, and hence have a smaller classical contribution (i.e., the unbound states). However, in the hydrodynamic approach, under high lateral electric field conditions, significant carrier heating can occur near the drain edge. Such energetic carriers tend to occupy bound states in the higher subbands and the unbound states. Hence, for high drain bias, the electron distribution near the drain edge calculated from HD 3-subband is more classical-like.
The Ia-Va characteristics from the HD simulations are presented in Figure 4. Either a temperature corrected van Dort's model or the 3-subband quantum mechanical model is used in the HD